Automata and finite order elements in the Nottingham group

Autor: Byszewski, Jakub, Cornelissen, Gunther, Tijsma, Djurre, Sub Fundamental Mathematics, Fundamental mathematics
Přispěvatelé: Sub Fundamental Mathematics, Fundamental mathematics
Jazyk: angličtina
Rok vydání: 2022
Předmět:
Zdroj: Journal of Algebra, 602, 484. Academic Press Inc.
ISSN: 0021-8693
Popis: The Nottingham group at 2 is the group of (formal) power series $t+a_2 t^2+ a_3 t^3+ \cdots$ in the variable $t$ with coefficients $a_i$ from the field with two elements, where the group operation is given by composition of power series. The depth of such a series is the largest $d\geq 1$ for which $a_2=\dots=a_d=0$. Only a handful of power series of finite order are explicitly known through a formula for their coefficients. We argue in this paper that it is advantageous to describe such series in closed computational form through automata, based on effective versions of proofs of Christol's theorem identifying algebraic and automatic series. Up to conjugation, there are only finitely many series $\sigma$ of order $2^n$ with fixed break sequence (i.e. the sequence of depths of $\sigma^{\circ 2^i}$). Starting from Witt vector or Carlitz module constructions, we give an explicit automaton-theoretic description of: (a) representatives up to conjugation for all series of order 4 with break sequence (1,m) for m
Comment: 49 pages, 18 figures; arxiv submission includes ancillary files: Mathematica notebooks (+pdf) containing examples and verifications;, a txt-file containing a MAGMA routine by A. Bridy and G. Cornelissen to compute the labelled graph structure of an automaton from an algebraic equation (using differential forms on algebraic curves); v2: small corrections
Databáze: OpenAIRE
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